COMPRESSED SPECTRUM SENSING IN THE PRESENCE OF INTERFERENCE:
COMPARISON OF SPARSE RECOVERY STRATEGIES
Eva Lagunas and Montse N´ajar
Department of Signal Theory and Communications,
Universitat Polit`ecnica de Catalunya (UPC), Barcelona, Spain
Email: {eva.lagunas,montse.najar}@upc.edu
ABSTRACT
Existing approaches to Compressive Sensing (CS) of sparse spectrum has thus far assumed models contaminated with noise (either bounded noise or Gaussian with known power). In practical Cognitive Radio (CR) networks, primary users must be detected even in the presence of low-regulated trans-missions from unlicensed systems, which cannot be taken into account in the CS model because of their non-regulated na-ture. In [1], the authors proposed an overcomplete dictio-nary that contains tuned spectral shapes of the primary user to sparsely represent the primary users’ spectral support, thus allowing all frequency location hypothesis to be jointly eval-uated in a global unified optimization framework. Extraction of the primary user frequency locations is then performed based on sparse signal recovery algorithms. Here, we com-pare different sparse reconstruction strategies and we show through simulation results the link between the interference rejection capabilities and the positive semidefinite character of the residual autocorrelation matrix.
Index Terms— Spectrum Sensing, Compressive Sensing, Interference Mitigation, Cognitive Radio
1. INTRODUCTION TO THE SPECTRUM SENSING PROBLEM
The basic idea of Cognitive Radio (CR) is spectral reusing or spectrum sharing, which allows the unlicensed users to com-municate over licensed spectrum when the licensed holders are not fully utilizing it [2]. In this context, the main goal of spectrum sensing is to decide whether a given frequency band is occupied by a primary user or not based on the observation of the received signal [3]. Let us denote x(t) the wideband signal representing the superposition of different primary sys-tems in a CR network. Signal x(t) is assumed to be sparse This work was partially supported by the Spanish Ministry of Science and Innovation (Ministerio de Ciencia e Innovaci´on) under project TEC2011-29006-C03-02 (GRE3N-LINK-MAC), by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM# (Grant agreement no. 318306), by the Catalan Government un-der grant 2009 SGR 891 and by the European Cooperation in Science and Technology under project COST Action IC0902.
multiband signal, i.e, a bandlimited, continuous-time, squared integrable signal that has all of its energy concentrated in a small number of disjoint frequency bands. In other words, its spectral support has Lebesgue measure small relative to the overall signal bandwidth [4]. Denoting the Fourier transform of x(t) as X(f ), the spectral support of the multiband signal x(t), denoted as F ⊂ [0, fmax], namely X(f ) = 0, f /∈ F ,
is restricted to have its signal energy distributed on no more than Nbdisjoint bands in F :
F =
Nb
[
i=1
[ai, bi) (1)
where [ai, bi), i = 1, . . . , Nb, represent the edges of each
band. The spectral occupancy Ω is defined as, Ω = λ(F )
fmax
0 ≤ Ω ≤ 1 (2)
where λ(F ) is the Lebesgue measure of the frequency set F , which, in this particular case, is equal toPNb
i=1(bi− ai). In
the spectrum sensing framework, the spectral support F is unknown but the total bandwidth under study is assumed to be sparse, i.e., Ω6 1.
The received signal y(t) consists on the superposition of multiple primary and secondary (interference) services. The non-regulated activity of the spectrum will be denoted as i(t) and henceforth is considered interference. The problem of determining whether a given frequency is occupied or not by a licensed radio can be modeled into a binary hypothesis test,
y(t) = (
i(t) + η(t) (H0)
x(t) + i(t) + η(t) (H1)
(3)
where (H0) stands for the absence of primary signal and (H1)
represents the presence of a primary signal in the frequency channel under study. η(t) is AWGN, N (0, ση2). The
user generally transmits at a lower rate than the primary in or-der to not disturb the quality of service of the primary user. Therefore, and as a baseline, it is assumed throughout this section that the interference has smaller bandwidth than that of the primary. The presence of unknown interference, which has not been addressed in conventional spectrum sensing pub-lications, adds additional challenges to the spectrum sensing problem. Note that information on primary users activity may potentially be used by the three relevant cognitive paradigms, namely interweave, underlay and overlay [5].
Let B denote the set of possible primary users’ frequency locations and let {ωm}
M −1
m=0 denote a grid that covers B.
As-suming that the true location of the primary users lie on the grid, the received signal y(t) can be rewritten as,
y(t) =
M −1
X
m=0
x(t, ωm) + i(t) + η(t) (4)
where x(t, ωm) denotes the analytic representation of the
pri-mary user corresponding to ωm, which is given by,
x(t, ωm) = am(t)ejωmt (5)
where am(t) indicates the complex envelope of the source at
ωm. Note that only a few am(t) are different from zero. For
the sake of simple notation, we are assuming linear propaga-tion channel with no distorpropaga-tion1.
2. COMPRESSIVE SAMPLING
The Compressive Sensing (CS) based candidate detector pre-sented by the authors in [1] makes the assumption that only a small number of primary users exists and therefore only few grid points are occupied, i.e., denoting K the number of pri-mary users present in y(t), K << M . As the value of M increases with respect to K, B is more and more sparse. CS theory states that a sparsely representable signal can be recon-structed using very few number of measurements [7]. In [1], the multi-coset (MC) sampling was proposed for compressive data acquisition. Given the received signal y(t), the MC sam-ples are obtained at the time instants,
ti(n) = (nL + ci)T (6)
where L > 0 is a suitable integer, i = 1, 2, . . . , κ and n ∈ Z. The set {ci} contains κ distinct integers chosen from
{0, 1, . . . , L − 1}. The MC sampling process can be viewed as a classical Nyquist sampling followed by a block that dis-cards all but κ samples in every block of L samples period-ically. Thus, a sequence or coset of equally-spaced samples is obtained for each ci. The period of each one of these
se-quences is equal to LT . Therefore, one possible implemen-tation consists of κ parallel ADCs, each working uniformly with period LT .
1The robustness in front of a frequency selective channel, instead of flat
fading, have been studied in [6]
The complete observation consists of a data record of Nf
blocks of κ non-uniform samples noted as yf. In order to
relate the acquired samples yf with the original
Nyquist-sampled signal, let us consider zf as the f -th block of L
uni-form Nyquist samples of y(t),
zf =y(tf1) . . . y(t f L) T (7) where tf
n = (f L + n)T . Thus, the relation between the
Nyquist samples and the sub-Nyquist samples is given by,
yf = Φzf (8)
where Φ is a matrix that randomly selects κ samples of zf
(κ < L). This matrix Φ is known as sampling matrix and is given by randomly selecting κ rows of the identity matrix IL.
3. CS-BASED SPECTRUM SENSING VIA SPECTRAL SHAPE FEATURE DETECTION
This section reviews the CS-based spectral shape detector proposed in [1]. It is assumed that the primary signal x(t) is unknown, i.e., the transmitted symbols are not known a priori. For such a case, it is appropriate to define a second-order based detector. To do so, the spectrum characteristics of the primary signals, which can be obtained by identifying its transmission technologies, are used as features. The basic strategy of the candidate detector is to compare the a priori known spectral shape of the primary user with the power spec-tral density of the received signal. To avoid the power specspec-tral density computation of the received signal, the comparison is made in terms of autocorrelation by means of a correlation matching. The diagram of the CR receiver is sketched in Fig. 1. Sub−Nyquist Sampling Correlation Estimation CS−based correlation matching
Fig. 1. Block diagram of the cognitive receiver
Firstly, taking advantage of the sparsity of the primary user’s signal spectrum, a sub-Nyquist sampling is used to overcome the problem of high sampling rate. Then, the com-pressed samples are processed in the autocorrelation estima-tion stage and finally, the correlaestima-tion-matching based spec-trum sensing is performed using a predetermined spectral shape, which has to be known a priori. Following the no-tation of (8), the sample autocorrelation matrix ˆRy ∈ Cκ×κ
For the purpose of the present work, let us define Rbas
base-band candidate autocorrelation of the primary system. The expression of Rb is linked to the PSD of the primary
sys-tem, which, for linearly modulated signals, depends mainly on the transmission rate and the modulation pulse. The afore-mentioned characteristics can be analytically extracted from physical layer standardization of primary services. In order to obtain the frequency location of each primary user, Rb is
modulated by a rank-one matrix formed by the steering fre-quency vector at the sensed frefre-quency ωmas follows,
Rc(ωm) =Rb e(ωm)eH(ωm) (10)
where denotes the elementwise product of two matrices and e(ωm) =1 ejωm . . . ej(L−1)ωm
T
is the steering vec-tor.
According to this notation, the corresponding model for the sample autocorrelation matrix defined in (9) is given by,
ˆ Ry = M −1 X m=0 γ(ωm)ΦRc(ωm)ΦH+ R (11)
where R represents the contribution of the
sub-Nyquist-sampled interference and noise autocorrelation matrices and γ(ωm) is the power level at frequency ωm.
The model in (11) can be conveniently rewritten into a sparse notation as follows,
ˆ
ry= kron(Φ, Φ)BSp + r= Ap + r (12)
where kron(·, ·) denotes the Kronecker product. Vector ˆry∈
κ2 × 1 is formed by the concatenation of the columns of
ˆ
Ry. From now on, to clarify notation, the concatenation of
columns will be denoted with the operator vec(·). Therefore, ˆry = vec( ˆRy). B contains the spectral information of the
pri-mary signals and is defined as diag(rb) where rb= vec(Rb).
Matrix S defines the frequency scanning grid, S =s(ω0) s(ω1) · · · s(ωM −1)
(13) where s(ωm) = vec(e(ωm)eH(ωm)). The variable r
en-compasses interference and noise contribution. Vector p = p(ω0) p(ω1) · · · p(ωM −1)
T
can be viewed as the out-put of an indicator function, whose elements different from zero correspond to the frequencies where the reference is present. Moreover, the values different from zero correspond to the power of each primary user that is present. This is,
p(ωm) =
(
γ(ωk) if ωm= ωk (H1)
0 otherwise (H0)
(14)
4. SPARSE RECONSTRUCTION STRATEGIES The previous sparse formulation begs for the use of well-known sparse reconstruction strategies. The aim of this pa-per is to compare different sparse reconstruction methods to recover the true primary users’ frequency location, i.e., the sparse vector p.
4.1. Conventional l1-norm minimization
Conventional l1-norm optimization can be applied to the
re-duced number of observations to recover the positions of pri-mary users, min p(ωm)≥0 kˆry− Apkl2+ λ kpkl1 (CONV) where kpkl 1 = P
m|p(ωm)| and λ is the regularization
pa-rameter that essentially controls the trade-off between the sparsity of the solution and its fidelity to the measurements. How to calculate the optimal λ is still an open problem. The l1-minimization is a convex problem and can be solved using
convex optimization techniques [8]. One important drawback of (CONV) is the difficulty on rejecting interference. The in-terference immunity is not achieved only using a dictionary of candidates because, although the primary signals’ spectral shape are assumed to be of different nature than that of the interference, they might not be orthogonal.
4.2. Weighted l1-norm minimization
Recently, the l1-norm has been shown to penalize large
coef-ficients to the detriment of smaller coefcoef-ficients [9]. Weighted l1-norm have been proposed to democratically penalize
nonzero entries. A novel weighted l1-norm minimization was
recently introduced by the authors in [1] to monitor the pri-mary user spectrum activity in CR. The proposed weights {wm}M −1m=0 were derived based on the positive-semidefinite
character of the residual error matrix R. Those weights were
given by the maximum eigenvalue of ˆR−1y (ΦRc(ωm)ΦH).
Therefore, all the hypothesis {ωm} M −1
m=0 can be evaluated in
a global unified convex optimization by solving the following problem,
min
p(ωm)≥0
kˆry− Apkl2+ λ kWpkl1 (P0)
where W is the diagonal matrix with the weights {wm} M −1 m=0
on the diagonal and zeros elsewhere.
4.3. Weighted and Constrained l1-norm minimization
It was mentioned that preserving the positive-semidefinite character of the residual error matrix R significantly
im-proves the interference rejection capabilities of the CS-based spectral shape detector. However, we will show on the simu-lations section that the weighted formulation of the 11-norm
minimization is not enough to not include the interference on the solution. For comparison purposes, let us consider the fol-lowing unweighted and weighted l1-norm minimization
prob-lems, min p(ωm)≥0 kˆry− Apkl 2+ λ kpkl1 s.t. ˆRy− M −1 X m=1 γ(ωm)ΦRc(ωm)ΦH 0 (P1) min p(ωm)≥0 kˆry− Apkl2+ λ kWpkl1 s.t. ˆRy− M −1 X m=1 γ(ωm)ΦRc(ωm)ΦH 0 (P2)
where a restriction on the positive semi-definite character of the residual error matrix has been imposed.
5. SIMULATION RESULTS
In this section, we evaluate the performance of the different sparsity-based reconstruction algorithms using synthesized data. The spectral band under scrutiny has bandwidth equal to fmax= 20 MHz. The size of the observation zf is L = 33
samples. The sampling rates of yf and zfare related through
the compression rate ρ = Lκ. To strictly focus on the per-formance behavior due to compression and remove the effect of insufficient data records, the size of the compressed ob-servations is forced to be the same for any compression rate. Therefore, we set M = 2Lδρ−1where δ is a constant (in the following results δ = 10). Thus, for a high compression rate, the estimator takes samples for a larger period of time. Note that the value of M determines the grid density of the spec-trum to be scanned. Increasing M makes the grid finer, but it also increases the computational complexity. On the other hand, making the grid too rough might introduce substantial bias into the estimates. The grid resolution for this section is ∆ω = 100 kHz. The l1-norm minimizations are solved with
the convex optimization program CVX [10].
To test the ability of the reconstruction techniques to prop-erly label licensed users, we have run 1000 Monte Carlo itera-tions (based on the noise randomness) of a scenario with one primary user in the presence of noise and interference. The interference is included as a 10 dB pure tone at normalized frequency 0.75, whereas the primary user is assumed to be a BPSK signal with a rectangular pulse shape and 8 samples
0 0.2 0.4 0.6 0.8 1 −20 −15 −10 −5 0 5 10 15 Norm. Frequency dB ← 8.51 dB ← 10.50 dB (a) 0 0.2 0.4 0.6 0.8 1 −20 −15 −10 −5 0 5 10 15 Norm. Frequency dB ← 10.09 dB ← 8.44 dB (b) 0 0.2 0.4 0.6 0.8 1 −20 −15 −10 −5 0 5 10 15 Norm. Frequency dB ← 8.90 dB ← −7.69 dB (c) 0 0.2 0.4 0.6 0.8 1 −20 −15 −10 −5 0 5 10 15 Norm. Frequency dB ← 9.41 dB ← −10.67 dB (d)
Fig. 2. CVX figures for ρ = 0.52: (a) CONV, (b) P0, (c) P1,
and (d) P2.
per symbol. The SNR of the desired user is 10 dB and its nor-malized carrier frequency is 0.25. The spectral occupancy for this particular example is Ω = 0.25 (the primary user is using one quarter of the available spectrum). According to Lan-dau’s lower bound [4], the minimum average sampling rate for most signals is given by the Nyquist rate multiplied by the spectral occupancy, which is often much lower than the corre-sponding Nyquist rate. For this particular example, Landau’s lower bound is telling us than 3 of every 4 Nyquist samples can be thrown away without affecting the signal reconstruc-tion. The results shown next consider λ = 1 and ρ = 0.52, which corresponds to κ = 17 and M = 1281.
Fig. 2 shows the average performance of the different CS-based reconstruction strategies presented in the previous sec-tion after 1000 Monte Carlo runs. Fig. 2(a) displays the con-ventional l1-norm minimization, which is not robust to the
strong interference. The use of the proposed weights helps in reducing the interference contribution, as shown in Fig. 2(b), but is not enough to discern between the primary user and the interference signal. The advantage of imposing positive semi-definite residual correlation matrix is evident from Figs. 2(c) and 2(d), where the difference between the peak corre-sponding to the true primary user and the peak correcorre-sponding to the interference has been significantly increased compared to Figs. 2(a) and 2(b). Moreover, we see from Fig. 2(d) a marked improvement over the unweighted l1-norm
recon-struction: the use of weights helps in reducing the interfer-ence level in 3dB.
We would like to confirm the advantages of the weighted and constrained l1-norm minimization (P2) over its
−9 −8 −7 −6 −5 −4 −3 −2 −1 0 0.2 0.4 0.6 0.8 1 Test Statistic Histogram: Primary Histogram: Interference PDF Approx. PDF Approx.
Fig. 3. Theoretical and empirical distributions of H1 and H0
for interference’s SNR = 7.5 dB and ρ = 0.52.
evaluate the probability of detection (Pd) versus SNR of the
interference for the same scenario as before but with the pri-mary user transmitting at SNR = 0dB.
Theoretical derivation of a detection threshold to meet the required Pf a requires the statistical distribution of both H1
and H0, which is a difficult task, especially for functions re-sulting from optimization problems. To find an approximate distribution of H1 and H0, we recorded 4.000 Monte Carlo simulations, keeping the peak value of the true primary user frequency location (H1) and keeping the peak value corre-sponding to the interference (H0). Fig. 3 shows the normal-ized distributions of H0 and H1 for a particular example with interference’s SNR = 7.5 dB and ρ = 0.52 solving (P2). The approximated probability density function computed using a kernel smoothing method is also plotted in Fig. 3 and it is seen that both distributions closely match the nonparametric approximation. In view of the results, we decided to adopt nonparametric density estimates for the computation of prob-ability figures.
Fig. 4 shows the Pdversus SNR results of the different
CS-based reconstruction strategies for a fixed probability of false alarm Pf a = 10−3 and ρ = 0.52. As predicted from Fig.
2, the weighted l1-norm minimization together with the
re-striction of keeping the positive semi-definite character of the residual error matrix provides the best detection capabilities. We see from Fig. 4 that solving (P2) provides a Pd = 0.99
for interference’s SNR = 7dB, while the other strategies fail in rejecting such strong interference.
6. CONCLUSION
This paper compares different CS-based reconstruction strate-gies for extraction of primary user frequency location in CR. Unlike other works, the proposed model considers the exis-tence of interferences emanating from low-regulated trans-missions, which cannot be taken into account in the CS model because of their non-regulated nature. From the results above, we conclude that keeping the positive semi-definite character of the residual correlation matrix together with forcing spar-sity is fundamental when dealing with CS models corrupted by unknown interference. −8 −6 −4 −2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 SNR Interf. Pd CONV P0 P1 P2
Fig. 4. Probability of detection versus SNR for Pf a = 10−3.
The scenario considers a BPSK primary signal (rectangular pulse shape, 8 samples per symbol) with SNR = 0dB, located at normalized frequency 0.25. There is a pure tone interfer-ence located at normalized frequency 0.75.
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